代写代考 DERMAN 2022.3

BACK TO THE CONTINUUM LIMIT OF MANY SMALL BINOMIAL STEPS LEAD TO:
ARITHMETIC BROWNIAN MOTION
4700: Intro to Fin Eng: Brownian Motion 26/33
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9.7 What is the transition probability distribution Px t;X T given that
X–x = T–t X–x2–X–x2 = 2T–t?
PnrNR = pPnrN–1R–1+1–pPnrN–1R
n r N R
We will use the recursion relation:
to find the relation between time n and time n + 1, and then let the time step become small and turn the relation into a PDE — -Planck/Forward Kolmogorov Equation for the evolution of probabilities– and solve it to find the distribution.
4700: Intro to Fin Eng: Brownian Motion 27/33
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Continuous probability distribution starting from (0,0) using Jarrow- of Binomial Model
P0 0;x t  Px t for convenient temporary notation
Conservation of Probability:
In discrete steps. Now go to the continuum limit to find and solve the pde for the continuous distribution:
The Fokker-Planck/Forward Kolmogorov Equation for evolution of probabilities:
position x+
P(x+,t-dt) 1/2
Pn – 1 r Pn – 1 r – 1
1/2 Let Px t be the probability density of being between x and x + dx at time t:
Then rewriting the recursion relation in terms of x and t rather than n and r:
Px+t = 0.5Px+t–t+0.5Px–t–t 4700: Intro to Fin Eng: Brownian Motion
P(x-,t-dt)
is the drift – deviation from mean 0
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Deriving the Fokker-Planck Equation for P(x,t) as t  0 Px+t = 0.5Px+t–t+0.5Px–t–t
Work out Taylor series:
 = –  – t to order t
Then divide by t
But in limit t  0 from Equations 9.5: – =  drift
This is the forward Fokker-Planck equation for Px t
+ P(x+e,t-dt) 1/2
 1/2 P(x-e,t-dt)
12P P P – – =
2 x2 x t
x 2x2 t
22 1PP P — - – – = 2x2 t x t t
and —- = 2 variance t
When you see a differential equation like this, it looks impenetrable. Understand that it really
In the continuum limit only  and 2 matter.
It’s a diffusion equation for x as a function of time t.
only represents the forward recursive binomial process in the limit of continuous motion.
4700: Intro to Fin Eng: Brownian Motion 29/33
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… Fokker- Planck Equation in the Continuum Limit
The pde for the distribution Px tthen becomes 2
12P P P - – =
2 x2 x t
In the continuum limit only  and 2 matter.
This is called the forward Fokker-Planck equation. The diffusion equation.
It just says that probability is conserved through time as the particle moves through the tree.
When you see a differential equation like this, understand that it really only represents the forward recursive binomial process in the limit of continuous motion.
Forward because it tells you how probability propagates forward. We showed above that the limit was chosen so that after time t,
x = t x2–x2 = 2t
The mean displacement is t and the mean variance of displacement around the mean is 2t.
This motion is called Arithmetic Brownian motion.
4700: Intro to Fin Eng: Brownian Motion 30/33
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Arithmetic Brownian Motion
The solution to this differential equation for t  0 is the normal distribution, subject to the initial condition of being at position x = 0 at time t = 0 .
12P P P
– – = PDE 2 x2 x t
1  x–t2
P0 0;x t = ——————-exp–——————— the probability density function (pdf)
22t  22t  x  Nt 2t
Prove Px t satisfies the PDE
Check that the integral over all x from – to  at any time t is always 1.
4700: Intro to Fin Eng: Brownian Motion
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Some exercises:
Provex = t x2–x2 = 2t.
Think about where the exponential isn’t very small.
When  or t get small, the curve gets higher and narrower, with area always equal to 1.
WhatdoesP(x,t)looklikewhent = 0?It’saDiracDeltafunction,widthzero,heightinfinity, area 1, that has all the probability concentrated at one point, x = 0.
4700: Intro to Fin Eng: Brownian Motion 32/33
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Delta Function as the Limit of a Normal Distribution with  = 0
Picture a Dirac delta function as the limit of N 2 with  = 0 zero drift and 2  0 0ifx0 
4700: Intro to Fin Eng: Brownian Motion
x=ifx = 0 and xdx = 1 –
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Picture a Dirac delta function x as the limit of N 2 with  = 0 zerodriftand20
0ifx0  x=ifx = 0 and xdx = 1
4700: Intro to Fin Eng: Brownian Motion
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Moment Generating Function for a Normal Distribution
A random variable X follows a normal distribution with mean  and variance 2 if its probability density function (pdf) is
212 px = Nx    = —————–e
1x – 2 – –  ———–
x  – 
We can obtain the moment generating function Ma by completing the square and integrating
over x, for all real a:
a + –a22
Ma = EeaX = e 2
Differentiating n times w.r.t a and setting a = 0, we obtain the nth moment about the origin:
Ma = E 1+aX+–a2X2+… = e
a  + — – a 2  2 2
M’a = EX+aX2+a2… = +a2e 2 etc 11
a + -a22 a + –a22 M”a = EX2+a… = +a22e 2 +2e 2
M’0 = EX =  M”0 = EX2 = +2 4700: Intro to Fin Eng: Brownian Motion
a + -a22
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Arithmetic vs Geometric Motion
An arithmetic series produces successive terms by adding. A geometric series produces successive terms by multiplication.
A random series of annually compounded returns 0.5, -0.5
If we take the additive average, the mean return is 0. This is arithmetic change.
But if these returns act on prices, starting with price 100, we get
100  1.5  0.5 = 100  0.75 = 75 and the return is not zero but rather -0.25. This is geometric because successive terms multiply.
4700: Intro to Fin Eng: Brownian Motion 16/29
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… Arithmetic Brownian Motion from the Central Limit Theorem
That the binomial distribution converges to a normal distribution also follows from the Central Limit Theorem, which says that the sum of N iid (independent identically distributed random
variables) with mean m and variance 2 approaches, as the sum becomes large, a normal distribution with mean Nm and variance N2.
We’ve already seen this from the binomial tree:
r–r2 = r2–r2 = variance = np1–p
4700: Intro to Fin Eng: Brownian Motion 17/29
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Stochastic Calculus of Brownian Motion
Why Stochastic Calculus?
Suppose, as an example, that you have an algorithmic trading strategy, a formula nS t, that involves buying shares whenever the stock goes down and selling shares whenever the stock goes up,
according to some formula. The rule is to sell nS tdt shares during the next instant of time dt when the stock price is S at time t, and S is random. If nS t is negative, that means we buy.
In that example, the cash available at the end is To nS tSerT – tdt and the total number of shares sold is worth STTo nS tdt at time T. These integrals are path-dependent, and we are
integrating a function of a stochastic variable S.
We need to be able to integrate over such Brownian motions in order to model and evaluate trading
strategies that involve holding variable quantities of stock nS t as S changes randomly, to take averages, variances, etc.
4700: Intro to Fin Eng: Brownian Motion 18/29
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10.5 Brownian Processes More Generally and Formally.
A Brownian process or a Wiener process.
The binomial tree in the limit t  0 is a particular representation of a more general and
abstractly defined stochastic process.
Definition
A Brownian process is a stochastic process Xt;t  0 with the properties
1. Every increment Xt + s – Xs is normally distributed with mean t and variance 2t, with  and  as fixed parameters;
2. For every t1  t2  …  tn the increments Xti – Xti – 1 are independent random variables with normal distributions as in 1. above.
3. X0 = 0 and the paths emanating from there are continuous. 1  x–t2
P0 0; x t = ——————-exp–——————— 22t  22t 
A standard Brownian motion or standard Wiener process Zthas  = 0 and 2 = 1.
2t 2t 4700: Intro to Fin Eng: Brownian Motion
1 x2 Px t = ———–exp–—-
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Some Properties
Show by integration for a standard Wiener process:
EZt = 0
EZt2 = varZ = t as we saw from the binomial distribution and its limit
For two independent Brownian motions X and Y: EXY = EXEY = 0 Another useful property is that EZtZs = mint s
Assume t  s . Then this trick is used a lot, separating times: EZtZs = EZt – ZsZs + Zs2
= EZt – ZsZs + EZs2
= 0 + s (first term is zero because independent increments) = mins t
The overlap in the processes gives variance proportional to the time that the processes overlap. Brownian motions have no autocorrelation.
4700: Intro to Fin Eng: Brownian Motion 20/29
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Illustration of EZtZs = mint s for the Binomial Tree Let s = 2t:
11 11 ZtZs =  t -2 t+-0 +– t –-2 t+-0 = 
4700: Intro to Fin Eng: Brownian Motion
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10.6 Stochastic Calculus (a Heuristic Treatment)
Calculus, differential and integral, lets you handle the motion of differentiable variables.
Brownian motions are not really differentiable because they jump around, continuously but abruptly. They are stochastic processes. Nevertheless one can integrate Brownian motions too, and treating them in this way is called stochastic calculus.
We’re interested in integration of random variables but we will often write derivatives as a short-hand for the inverse process of integration.
d f  x  y
Thus in ordinary calculus, – = x + 3 also means fy – f0 = x + 3dx
You can think of a Brownian motion as the limit of a discrete random walk process on a binomial tree with n periods over time t, with the limit taken as n   for fixed t.
The paths are non-smooth and obviously not differentiable in the normal sense, but they are continuous and therefore integration can make sense. Differentiation makes things more spiky, but integration averages over spikes.
So, in order to approach integration, let’s look at a binomial tree with n periods and discover some interesting and important properties of such paths and integrals of functions over them.
4700: Intro to Fin Eng: Brownian Motion 22/29
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A Random Walk for Z(t) Has Known Finite Quadratic Variation
standard 0 1 t+ t
Binomial process for random variable Z
Quadratic Variation Q( ): The square of the changes along the path
Q0t = Zi–Zi–1 =  t = nt = n-t n
i=1 i=1 This will hold even in the limit as n   for fixed t.
The absolute variation is infinite: n  t = —- t = ——–   as t  0
t t 4700: Intro to Fin Eng: Brownian Motion
0 1 2 3 i… n n levels, each t apart, total time t
random path
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In Contrast, An Ordinary Differential Variable Has Zero Quadratic Variation. A Brownian Motion is Not Differentiable.
• For a differentiable function Xt, X is finite, and the quadratic variation is zero. t
Xt + dt – Xt = Xdt + … higher order terms in dt t
Q0t =  Xi–Xi–12   t nt2 = n- 0 as n
Q0 t = t means that Z(t) is not a differentiable function of time, though it is continuous.
—–  –  ——–   as dt  0
In finance we are interested in integrating or summing over stochastic paths to see what happens to the profit and loss as we trade, or to understand the statistics of random prices.
Thus we will want to look at the stochastic Ito integral.
4700: Intro to Fin Eng: Brownian Motion 24/29
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10.7 The Ito Process
An Ito process describes a random path Xt generated with a variable volatility:
Xt = X0+t0sds+t0sdZs Eq.10.9
As dZs is a random Wiener process, this generates many paths through time.
The differential form that corresponds to the integral which is the standard way of writing it is
dXt = tdt + tdZt
We must explain what we mean by the integral t0sdZs in the equation above.
And if we define a random path then we want to define an integral along the path.
4700: Intro to Fin Eng: Brownian Motion 25/29
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0 n  i = 1
ft is adapted or non-anticipative: it depends only on the path history of Zt up to time t.
In finance, the function f  usually reflects what we trade, e.g. how many shares S t we buy in a trading strategy when the stock is at price S at time t . Then the subsequent change in Z(t) reflects what happens to the price S after that.
Let’s see what an Ito integral looks like in a particular case. Let’s integrate Zt by evaluating Tn
ZtdZt = lim  Zti – 1Zti – Zti – 1 0 n  i = 1
where Zt is a standard Wiener process. Let’s take n = 3 first as a simple example.
For ordinary variables,
4700: Intro to Fin Eng:
ZT2–Z02 ZtdZt = ———————————-
1 2 The Chain Rule ZdZ = -dZ
ftdZt  lim  fti – 1Zti – Zti – 1 Eq.10.10
2 Brownian Motion
forward differential
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… Now Let’s Look at the Binomial Sum for n = 3 out to Time t  Zti – 1Zti – Zti – 1
3 Zti – 1 + Zti – Zti – Zti – 1
————————————————————————————-Zti – Zti – 1 2
rearrange terms to get squares which we know how to handle via defn of Brownian motion quadratic variation.
12212 – Z ti–Z ti–1 –-Zti–Zti–1 22
i=1 12212
= –Z3 – Z0  – -Zti – Zti – 1 22
1221t = -Z3 – Z0  – – 3- 223
dZ2 =dt=t/3
Zti – 1Zti – Zti – 1 = -Z3 – Z0  – -t 22
It’s as though ZdZ = –dZ  – – or in differential form dZt
= 2ZtdZt + dt. Modified Chain Rule. We can generalize this as follows for arbitrary n periods covering time t.
4700: Intro to Fin Eng: Brownian Motion 27/29
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… The (Forward) for n > 3
T ZtdZt = lim  Zti – 1Zti – Zti – 1 0 n  i = 1
n Zti – 1 + Zti – Zti – Zti – 1 ———————————————————————————–Zti – Zti – 1
+Zt Zt –Zt i–1 i i i–1
= –  Zt
2 –Zt –Zt  
1 n  2 2 = — –    Z  t  – Z  t
  –  Z  t  – Z  t
2  i i–1 i i–1 
= – Z tn–Z 0–
dt ZtdZt = –Z T – Z 0 – T
In the limit of an infinitely fine grid over time t, with nt = T as n  , we obtain
4700: Intro to Fin Eng: Brownian Motion 28/29
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2ZtdZt = – T + ZT2 – Z02 Eq.10.11
This only has real meaning as an integral, but we can write the symbolic differential form that represents Equation 10.11:
2ZtdZt = dZ2t – dt or dZ2t = 2ZtdZt + dt
df = dZ+- dt
The extra dt term comes from the Brownian motion/quadratic variation of moves ~ dt.
We will often (always) write equations like Equations 10.12, which look like differential calculus, but they are really a short way of specifying what happens when you integrate. They describe the
stochastic process of the function Z2t
Equation 10.12 is an instance of Ito’s Lemma for Wiener processes. It is a version of the chain rule
for differentiation, but for stochastic processes.
4700: Intro to Fin Eng: Brownian Motion 29/29
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11.6 Ito’s Lemma:
A Statement About How To Integrate Functions Of A Stochastic Variable dXt = tdt + tdZt
2t 2 fX t + ————
dX2t = 2tdt dXdt = Odt3/2 = 0 dt2 = 0
Box algebra for a standard Wiener process keeping leading orders of dt in the Taylor expansion:
dfX t =
A heuristic proof follows from doing the Taylor expansion of fX t up to second order in X
and t, keeping all terms of order dt, and using the rules
fX t dt +
fX tdX Eq.11.8
4700: Intro to Fin Eng: Brownian Motion 18/27
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11.7 Geometric Brownian Motion for Stock Prices
Log returns Xt are modeled as normal. Stock prices are modeled as exponentiated returns. St = S eXt lnSt = Xt
This means returns can be positive or negative, normally distributed, and the exponentiated return is always positive, so the stock price never goes negative. (The original application of Brownian motion to stocks by Bachelier used ABM and allowed stock prices to go negative.)
Calibration of the Drift of the Return: suppose we observe that the stock price compounds at an average rate  so that ESt = S0et with volatility of returns .
How must we choose X(t) to grow so that S(t) increases at rate ?
X is an arithmetic Brownian motion for returns. so that dX = something dt + dZ.
What is the required drift something? It is not exactly .
What happens when you exponentiate an arithmetic Brownian motion? Well, we’ll see that if
the ABM has zero drift, the GBM has a greater positive drift. Let’s look at it binomially first.
4700: Intro to Fin Eng: Brownian Motion 19/27
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11.7.1 From Normal/Arithmetic to Lognormal/Geometric Binomially
WehaveearliermodeledthebinomialdistributionofanArithmeticrandomvariableX = lnS
as involving infinitesimal increments t  
1 p = — 2
u = t+ t
1 lnS+t+ t p = —
d = t– t
lnS+t– t
Repeated —- times, we showed this converges to a normal distribution Nt  t with drift
Now let’s see what happens when we exponentiate the arithmetic Brownian motion.
4700: Intro to Fin Eng: Brownian Motion 20/27
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ES = –S  e
= Set1 + -  Sete 2 2
lnS+t+ t
lnS+t– t 
Sexpt+ t
t+ t t– t te t+e– t + e  = Se ———————————————–
1 2  t expt+ t1+ t+– te
Sexpt– t t
 1 2  t
expt– t1– t+- te 2
2t 2t –
 2  + - t
The stock on average grows at an exponential rate  + —–
If we want the stock to grow at average rate  then the lnstock must grow at  – 2  2 Now let’s look at this more elegantly using Ito’s Lemma
4700: Intro to Fin Eng: Brownian Motion 21/27
COPYRIGHT EMANUEL DERMAN 2022.3

Using Ito’s Lemma for Geometric Brownian Motion
Assume dXt =  – - dt + dZt
S = fX t = S e
0 t X X2
S S 1S 2 dSt = dt+ dX+-  dt
t X 2X2
22 2 dSt = StdXt + St-dt = St  – —– dt + dZt + St-dt
S = 0 S = S,  S = S geometric
222 dSt = Stdt + dZt
———— = dt + dZt
take Z averages dS = Sdt
St = S0expt
If lnS has drift  – 2  2 then S has exponential growth rate .
Ito’s Lemma is really nothing more than analogous to a nonlinear function Taylor- expanded on a binomial tree.
4700: Intro to Fin Eng: Brownian Motion 22/27
COPYRIGHT EMANUEL DERMAN 2022.3

Summarizing Geometric Brownian Motion for the Stock Price
Stochastic Differential Equation
———— = dt+dZt or dSt = Stdt+StdZt
dlnSt =  – —–dt + dZt
2 St  2
ln———-

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